Subtree Number versus Wiener Index

The subtree number N(G) of a graph G is defined as the number of non-empty subtrees of G. The Wiener index W(G) of a graph G is defined as the sum of distances over all pairs of vertices in G. It has been noted that, for many families of trees and graphs, the graphs that achieve the largest number of subtrees are exactly those that attain the smallest Wiener index, and vice versa. Consequently, it is often said that the subtree number and the Wiener index have a ”negative” correlation. In this paper, we show that, except for extremal graphs, this ”negative” correlation does not generally hold. In particular, for every n ≥ 14, we construct a pair of unicyclic graphs G and H, each having n vertices and identical degree sequences, such that W(G) < W(H) and N(G) < N(H). Furthermore, our construction shows that both differences, N(H) − N(G) and W(H) − W(G), grow unbounded as n increases.